Some definitions of interest.
linorder	Def Linorder(T;x,y.R(x;y)) == Order(T;x,y.R(x;y)) & Connex(T;x,y.R(x;y))
	Thm* T:Type, R:(TTProp). Linorder(T;x,y.R(x,y))  Prop
strict_part	Def strict_part(x,y.R(x;y);a;b) == R(a;b) & R(b;a)
	Thm* T:Type, R:(TTProp), a,b:T. strict_part(x,y.R(x,y);a;b)  Prop
not	Def A == A  False
	Thm* A:Prop. (A)  Prop

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